[Paper Review] Existence of weak solutions to multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman models for stratified tumor growth with chemotaxis and general source terms

Patrik Knopf, Andrea Signori|arXiv (Cornell University)|Jan 1, 2022

Mathematical Biology Tumor Growth97 references25 citations

TL;DR

This paper establishes the existence of global weak solutions for a multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman system modeling stratified tumor growth with chemotaxis and general source terms. The model incorporates multiple cell types and chemical species (e.g., oxygen, nutrients), uses phase field functions to represent tumor and healthy cells, and couples Cahn–Hilliard equations for tumor evolution with reaction-diffusion equations for nutrients. The key result is the rigorous proof of existence of weak solutions for both the Brinkman and Darcy formulations via a vanishing viscosity limit approach.

ABSTRACT

We investigate a multiphase Cahn–Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn–Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy’s law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn–Hilliard–Brinkman model. After that, we show that such weak solutions of this system converge to a weak solution of the multiphase Cahn–Hilliard–Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn–Hilliard–Darcy system is also established.

Motivation & Objective

• To develop a multiphase mathematical model for tumor growth that captures stratified tissue layers (proliferating, quiescent, necrotic) more accurately than classical two-phase models.
• To incorporate chemotaxis, nutrient consumption, and general source terms in a diffuse interface framework.
• To rigorously establish the existence of global weak solutions for both the Cahn–Hilliard–Brinkman and Cahn–Hilliard–Darcy systems.
• To demonstrate the convergence of weak solutions of the Brinkman model to those of the Darcy model as viscosity tends to zero.
• To provide a theoretical foundation for patient-specific tumor growth simulations with complex tissue heterogeneity.

Proposed method

• Formulates a multiphase Cahn–Hilliard system with L tumor cell types and M chemical species (e.g., oxygen, nutrients), using phase field functions ϕ and σ.
• Couples the Cahn–Hilliard equation for tumor evolution with a reaction-diffusion equation for nutrient dynamics.
• Uses volume-averaged velocity v governed by either Brinkman’s law (with viscosity) or Darcy’s law (viscosity-free), enabling fluid-like and porous medium-like tumor dynamics.
• Employs a variational formulation with test functions in Sobolev spaces to derive weak solutions.
• Applies a vanishing viscosity limit technique: constructs solutions for the Brinkman model and passes to the limit as viscosity →0 to recover solutions for the Darcy model.
• Relies on compactness arguments, weak convergence, and a priori estimates to pass to the limit in nonlinear terms and recover the initial conditions.

Experimental results

Research questions

• RQ1 Can a multiphase Cahn–Hilliard model accurately describe stratified tumor growth with multiple cell types and chemical species?
• RQ2 Does the Cahn–Hilliard–Brinkman system admit global weak solutions with general source terms and chemotaxis?
• RQ3 Can weak solutions of the Cahn–Hilliard–Brinkman model converge to weak solutions of the Cahn–Hilliard–Darcy model as viscosity vanishes?
• RQ4 What is the mathematical justification for using Darcy’s law in tumor growth models when viscosity is negligible?
• RQ5 How can general source terms and chemotactic effects be consistently incorporated into a diffuse interface framework for tumor dynamics?

Key findings

• The existence of a global weak solution is rigorously established for the multiphase Cahn–Hilliard–Brinkman system with general source terms and chemotaxis.
• Weak solutions of the Cahn–Hilliard–Brinkman system converge to weak solutions of the Cahn–Hilliard–Darcy system as the viscosities tend to zero in a suitable sense.
• The limit process preserves the structure of the system, including the divergence-free condition for velocity and the interface dynamics governed by the Cahn–Hilliard equation.
• The initial conditions are preserved in the limit, ensuring that the solution satisfies ϕ|t=0 = ϕ₀ and σ|t=0 = σ₀ in the weak sense.
• The pressure field is shown to satisfy p|Σ = 0 almost everywhere on the boundary, confirming consistency with the Darcy-type momentum equation.
• The method provides a robust analytical framework for extending diffuse interface models to complex, multiphase tumor microenvironments with realistic biological features.